Sanaa Khan K1306336
FACULTY OF SCIENCE, ENGINEERING AND COMPUTING
School of Computer Science and
Mathematics
BSc (Hons) DEGREE
IN
Financial Mathematics with Business Management
Name: Sanaa Khan
ID Number: K1306336
Project Title: Valuation of Convertible Bonds
Date: 11/04/16
Supervisor: Luluwah Al-Fagih
WARRANTY STATEMENT
This is a student project. Therefore, neither the student nor Kingston University
makes any warranty, express or implied, as to the accuracy of the data or conclusion
of the work performed in the project and will not be held responsible for any
consequences arising out of any inaccuracies or omissions therein.
Abstract
In this paper, we will be discussing methods of pricing a European style convertible bond
(CB), i.e. where conversion can only take place at maturity. Pricing methods include using
the Black-Scholes model to price the bond by splitting components to help simplify the
procedure. Furthermore, contract features will be looked upon, to give a better perspective
as to what is said between the issuer and the bondholder, as well as how the CB is formed
and the features within it. The paper will also be looking at the analysis of price sensitivities
and how different features affect the price of a CB and the impact they have on a portfolio
containing a CB.
Sanaa Khan K1306336
Contents
Abstract i
1. Introduction 1
2. Payoff Profiles 4
2.1. Notation 4
2.2. Bondholders Perspective 5
2.3. Bond Issuers Perspective 5
2.4. Payoff 6
2.5. A Zero-Sum Game 7
2.5.1. Example 7
3. Contract Features of a Convertible Bond 9
3.1. Convertible Bond Financing 9
3.2. Maturity 9
3.3. Principle 9
3.4. Conversion Ratio 10
3.5. Call Provisions 10
3.6. Put Provisions 10
3.7. Coupon Payments 11
3.8. Refix Clause 11
3.9. Other Non-Standard Clauses 12
3.10. Termination 12
Sanaa Khan K1306336
4. Properties of a Convertible Bond 13
4.1. Conversion Price 13
4.2. Parity 13
4.3. Premium to Parity 14
4.4. Investment Premium 14
4.5. Bond Floor 15
4.6. Price Sensitivities 16
4.7. Upper and Lower Bounds 19
5. Pricing Methods 21
5.1. Mathematical Background 21
5.2. Monte Carlo Simulation 21
5.3. Lattice based Method 22
5.4. Reduced Form Approach 22
5.5. Tsiveriotis-Fernandes Method 23
5.6. Black-Scholes Method 23
6. Black-Scholes Model 26
6.1. The Bond Price 26
6.2. Example 29
6.3. Margrabe Formula 31
7. Conclusion 34
8. References 35
1
1. Introduction Convertible bonds (CB) were first used during the 1960s. Convertible bonds are hybrid
securities; they use both equity and debt. A convertible bond is a bond such that the holder
of the bond; that being the investor is able to convert it into cash or equity when they feel it
would be beneficial to them [7 - pg 58]. Ingersolls (1977) research suggests that the general
valuation procedure would be to set up the price of the convertible and equate it to the
maximum value of a straight bond, or the value it holds within the common stock (after
conversion) given that at some point in the near future. The value found from this, would
then be discounted back to the present value. Yan, Yi, Yang and Liang (2015) state they wish
to keep hold of the bond, in which case they will receive interest payments; or they could
convert it into the companys stocks. The bondholder would ideally pick a strategy in which
they would be able to maximise the CB value.
The issuers of convertible bonds are usually smaller firms. Smaller firms who are looking into
getting finances. The reason for this is because smaller firms are not as well-known and need
financing when their credit is low [20]. It is found that when a weaker firm wishes to issue a
CB, it shows they have faith in their project. This enhances their chances of gaining investors
for their company. However, a larger firm would not need to issue convertible bonds as they
would easily be able to get funding and or loans as they are more known within the industry.
If a larger firm wanted to issue a bond, they would not have enough buyers.
The motivation behind the smaller firms issuing the convertible bonds is due to the fact they
lack stable credit histories. This means they would have to pay higher interest payments;
also known as coupons - to their debt holders. The size of a firm usually is a reason as to why
there is an issuance of convertible bonds [12]. Firm size is associated with bankruptcy costs;
since smaller firms are more vulnerable to failure and are risk averse. Smaller firms face
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2
higher degree information asymmetry, this could increase the cost of the debt. It could also
lead to having more restrictive contracts also known as covenants, had they wanted to
issue a straight bond. This is a reason why larger firms just offer straight bonds. A convertible
bond is more flexible the way it works, matters are stated within the contract, as well as
being set out if the firm breaks the contract they (the bondholder) will receive a premium
[12]. The motivation behind issuing CBs is the fact firms will have interest rate-cost savings,
in comparison to issuing straight corporate1 bonds [20].
Another reason why firms issue CBs is to ensure the investor has no entitlement in the
running of the business. This would mean having the ability to vote for the directors that
would only be in control of the common stockholders. This makes it attractive to firms, as
they know their positions will not be endangered nor questioned. Kwok (2014) suggests that
convertible bonds are chosen by firms over straight bonds due to the lower coupon rate.
CBs have a callable feature which means it can be redeemed by the issuer prior to the
contractual date, this paper will follow a European styled CB. At this point, a price in the
form of a penalty, would be paid to the bondholder, as the company is forcing them to
either convert or surrender the bond [ref 7 page 58].
Owning a convertible bond is like playing a game. The bondholder is allowed to convert the
bond when they see it is beneficial for them. Suppose the bondholder converts before the
call date set within the covenant; it would mean the shareholders were not able to call the
bond when they thought it would be beneficial for them [7]. According to Yan, Yi, Yang and
Liang (2015) when the coupon rate is bounded above by the interest rate multiplied by the
strike price, that is when the bondholder will convert the CB. The conversion for the issuer
will take place when the coupon rate is lower than the dividend rate multiplied by the strike
1 Information Asymmetry: a party within a transaction has more information on the other party that
they are dealing with. Due to this, a party is likely to take advantage of the other partys lack of knowledge.
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3
price; though this paper will not be discussing dividends used within CBs. The contract is
terminated when the coupon rate lies in between the two bounds, at that point both parties
will terminate the contract.
The bondholder will be receiving coupon payments, over the life of the CB, up until the
contract has reached its expiry (maturity). Prior to maturity, the bondholder has the right to
convert their bond into the companys shares. Close to the end of the contract the company
have the right to call the bond back and force the bondholder to capitulate the bond to the
company.
In this paper, we will be discussing methods of pricing a European style convertible bond
(CB), i.e. where conversion can only take place at maturity. Pricing methods include using
the Black-Scholes model to price the bond by splitting components to help simplify the
procedure. Furthermore, contract features will be looked upon, to give a better perspective
as to what is said between the issuer and the bondholder, as well as how the CB is formed
and the features within it. The paper will also be looking at the analysis of price sensitivities
and how different features affect the price of a CB and the impact they have on a portfolio
containing a CB.
Sanaa Khan K1306336
4
2. Payoff Profiles
A payoff is what is received by the bondholder during the lifetime of the bond. The
bondholder has two options, (i) to receive the face value, or (ii) the share price multiplied by
the conversion ratio the one with the greater value is what the bondholder will receive.
First we introduce some notation that will be used throughout the paper.
2.1 Notation
Face value
Conversion ratio
Share price at time
Coupon payment
Maturity
Conversion price
Strike price
Bond floor
